Need help filling out your brackets? Watch these free videos from Tim Chartier

Chartier_MathStill rushing to fill out your brackets for the NCAA tournament? This free online course from mathematician Tim Chartier, author of Math Bytes, might help.

In this course, you will learn three popular rating methods two of which are also used by the Bowl Championship Series, the organization that determines which college football teams are invited to which bowl games. The first method is simple winning percentage. The other two methods are the Colley Method and the Massey Method, each of which computes a ranking by solving a system of linear equations. We also learn how to adapt the methods to take late season momentum into account. This allows you to create your very own mathematically-produced brackets for March Madness by writing your own code or using the software provided with this course.

From this course, you will learn math driven methods that have led Dr. Chartier and his students to place in the top 97% of 4.6 million brackets submitted to ESPN!

Explore Tim Chartier’s March MATHness lectures:

Pi Day: Where did π come from anyway?

This article is extracted from Joseph Mazur’s fascinating history of mathematical notation, Enlightening Symbols. For more Pi Day features from Princeton University Press, please click here.

 


 

k10204[1]When one sees π in an equation, the savvy reader automatically knows that something circular is lurking behind. So the symbol (a relatively modern one, of course) does not fool the mathematician who is familiar with its many disguises that unintentionally drag along in the mind to play into imagination long after the symbol was read.

Here is another disguise of π: Consider a river flowing in uniformly erodible sand under the influence of a gentle slope. Theory predicts that over time the river’s actual length divided by the straight-line distance between its beginning and end will tend toward π. If you guessed that the circle might be a cause, you would be right.

The physicist Eugene Wigner gives an apt story in his celebrated essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” A statistician tries to explain the meaning of the symbols in a reprint about population trends that used the Gaussian distribution. “And what is this symbol here?” the friend asked.

“Oh,” said the statistician. “This is pi.”

“What is that?”

“The ratio of the circumference of the circle to its diameter.”

“Well, now, surely the population has nothing to do with the circumference of the circle.”

Wigner’s point in telling this story is to show us that mathematical concepts turn up in surprisingly unexpected circumstances such as river lengths and population trends. Of course, he was more concerned with understanding the reasons for the unexpected connections between mathematics and the physical world, but his story also points to the question of why such concepts turn up in unexpected ways within pure mathematics itself.

 

The Good Symbol

The first appearance of the symbol π came in 1706. William Jones (how many of us have ever heard of him?) used the Greek letter π to denote the ratio of the circumference to the diameter of a circle. How simple. “No lengthy introduction prepares the reader for the bringing upon the stage of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded.” But for the next thirty years, it was not used again until Euler used it in his correspondence with Stirling.

We could accuse π of not being a real symbol. It is, after all, just the first letter of the word “periphery.” True, but like i, it evokes notions that might not surface with symbols carrying too much baggage. Certain questions such as “what is ii?” might pass our thoughts without a contemplating pause. Pure mathematics asks such questions because it is not just engaged with symbolic definitions and rules, but with how far the boundaries can be pushed by asking questions that everyday words could ignore. You might think that ii makes no sense, that it’s nothing at all, or maybe a complex number. Surprise: it turns out to be a real number!

It seems that number has a far broader meaning than it once had when we first started counting sheep in the meadow. We have extended the idea to include collections of conceptual things that include the usual members of the number family that still obey the rules of numerical operations. Like many of the words we use, number has a far broader meaning than it once had.

Ernst Mach mused:

Think only of the so-called imaginary quantities with which mathematicians long operated, and from which they even obtained important results ere they were in a position to assign to them a perfectly determinate and withal visualizable meaning.

It is not the job of mathematics to stick with earthly relevance. Yet the world seems to eventually pick up on mathematics abstractions and generalizations and apply them to something relevant to Earth’s existence. Almost a whole century passed with mathematicians using imaginary exponents while a new concept germinated. And then, from the symbol i that once stood for that one-time peculiar abhorrence √−1, there emerged a new notion: that magnitude, direction, rotation may be embodied in the symbol itself. It is as if symbols have some intelligence of their own.

What is good mathematical notation? As it is with most excellent questions, the answer is not so simple. Whatever a symbol is, it must function as a revealer of patterns, a pointer to generalizations. It must have an intelligence of its own, or at least it must support our own intelligence and help us think for ourselves. It must be an indicator of things to come, a signaler of fresh thoughts, a clarifier of puzzling concepts, a help to overcome the mental fatigues of confusion that would otherwise come from rhetoric or shorthand. It must be a guide to our own intelligence. Here is Mach again:

In algebra we perform, as far as possible, all numerical operations which are identical in form once for all, so that only a remnant of work is left for the individual case. The use of the signs of algebra and analysis, which are merely symbols of operations to be performed, is due to the observation that we can materially disburden the mind in this way and spare its powers for more important and more difficult duties, by imposing all mechanical operations upon the hand.

The student of mathematics often finds it hard to throw off the uncomfortable feeling that his science, in the person of his pencil, surpasses him in intelligence—an impression which the great Euler confessed he often could not get rid of.

A single symbol can tell a whole story.

There was no single moment when xn was first used to indicate the nth power of x. A half century separated Bombelli’s , from Descartes’s xn. It may seem like a clear-cut idea to us, but the idea of symbolically labeling the number of copies of x in the product was a huge step forward. The reader no longer had to count the number of x’s, which paused contemplation, interrupted the smoothness of reading, and hindered any broad insights of associations and similarities that could extend ideas. The laws xnxm = xn+m and (xn)m = xnm, where n and m are integers, were almost immediately suggested from the indexing symbol. Not far behind was the idea to let x½ denote √x, inspired by extending the law xnxm = xn+m to include fractions, so x½ x½  = x1.

Further speculation on what nx might be would surely have inspired questions such as what x might be for a given y in an equation such as y = 10x. Answer that and we would have a way of performing multiplication by addition. But Napier, the inventor of logarithms, already knew the answer long before mathematics had any symbols at all!

Symbols acquire meanings that they originally didn’t have. But symbolic representation has, likewise, the disadvantage that the object represented is very easily lost sight of, and that operations are continued with the symbols to which frequently no object whatever corresponds.

Ernst Mach once again:

A symbolical representation of a method of calculation has the same significance for a mathematician as a model or a visualisable working hypothesis has for a physicist. The symbol, the model, the hypothesis runs parallel with the thing to be represented. But the parallelism may extend farther, or be extended farther, than was originally intended on the adoption of the symbol. Since the thing represented and the device representing are after all different, what would be concealed in the one is apparent in the other.

Mirror, Mirror Book Giveaway!

simon

We are doing a book giveaway to celebrate the upcoming book birthday of Simon Blackburn’s Mirror, Mirror: The Uses and Abuses of Self-Love on March 26th .

Three lucky readers will each win one cloth copy of Mirror, Mirror.

How to enter? There are numerous ways to enter, including liking the Princeton University Press Facebook page, emailing us at blog@press.princeton.edu, tweeting about the giveaway or following at @PrincetonUPress, or pinning the author’s book selfie (above) on Pinterest. Just follow the steps in the Rafflecopter box below. The winners will be selected on Friday, March 21st.

a Rafflecopter giveaway

Top Tips for 2014 March Madness Brackets from Tim Chartier

Chartier_MathWith a $1 billion dollar payday on the line, we predict there will be more people filling out March Madness brackets this year than ever before, so it isn’t surprising that everyone is looking to mathematician Tim Chartier for tips and tricks on how to pick the winners. Tim has been using math to fill out March Madness brackets with his students for years and his new book Math Bytes will have an entire section devoted to best tips and tricks. In the meantime, we invite you to check out these tips from an interview at iCrunchData News.

ICrunchData: What are a few variables that are used that are out of the ordinary?

Chartier: “In terms of past years, it helps if you look at scores in buckets. For instance, you decide close games are within 3 points and count those as ties. Medium wins are 4 to 10 points and could as 6 points and anything bigger is an 11 point win. That’s worked really well in some cases and reduces some of the noise of scores.”

“Here is another that comes out of our most current research. This year’s tournament will enable us to test it in brackets. We tried it on conference tournaments and it had good success. We use statistics (specifically Dean Oliver’s 4 Factors) and look at that as a point, in this case in 4D space. Then we find another team that has a point in the fourth dimension closest to that team’s point. This means they play similarly. Suddenly, we can begin to look at who similar teams win and lose against.”

Congratulations to Howard Wainer for winning the 2014 AERA Division D Significant Contribution to Educational Measurement and Research Methodology Award from the American Educational Research Association

j9529[1]Uneducated Guesses: Using Evidence to Uncover Misguided Education Policies by Howard Wainer is winner of the 2014 AERA Division D Significant Contribution to Educational Measurement and Research Methodology Award, American Educational Research Association. The award “recognizes one publication that represents a significant conceptual advancement in theory and practice of educational measurement and/or research methodology. This year’s award recognizes such a publication in the area of Quantitative Methods and Statistical Theory. The evaluation criteria are quality, originality, and potential impact.”
In their citation for the award, the commitee notes, “In his book, Wainer describes, evaluates, and illustrates complex statistical reasoning embedded in a wide range of important educational policies in ways that are easily accessible and penetrable to non-technical people. This collective work evaluates the relationship between sophisticated statistical and psychometric machinery and challenges the educational policies and practices that have far-reaching impact on our society at large. Wainer’s thought-provoking writing regarding the perils of misusing quantitative measurement outcomes represents a significant contribution that is likely to shape educational research and practice for many generations.”

Pi Day Recipe: Brandy Alexander Pie from Cooking for Crowds

This recipe is presented as part of our Pi Day celebration. For more Pi Day features from Princeton University Press, please click here.


Brandy Alexander Pie

This pie is as sweet and delicious as the drink for which it is named, and a great deal less alcoholic. It is light and fluffy, but very filling.

6 12 20 50
unflavored gelatin envelopes 1 2 4 8
cold water ½ c 1 c 2 c 4 c
granulated sugar ⅔ c 1⅓ c 2⅔ c 2 lbs
salt ⅛ tsp ¼ tsp ½ tsp 1 tsp
eggs, separated 3 6 12 24
Cognac ¼ c ½ c 1 c 2 c
Grand Marnieror ¼ c ½ c 1 c 2 c
creme de cacao ¼ c ½ c 1 c 2 c
heavy cream 2 c 4 c 4 pts 8 pts
graham cracker crust 1 2 4 8
Garnish
4-ounce bars semisweet chocolate 1 2 3
heavy cream 1 c 2 c 3½ c 6 c

Sprinkle the gelatin over the cold water in a saucepan. Add ⅓ cup [⅔ cup, 1⅓ cups, 2⅔ cups] of the sugar, the salt, and egg yolks. Stir to blend, then heat over low heat, stirring, until the gelatin dissolves and the mixture thickens. Do not boil. Remove from the heat and stir in the Cognac and Grand Marnier (or creme de cacao). Chill in the refrigerator until the mixture mounds slightly and is thick.

Beat the egg whites until stiff (use a portable electric mixer in a large kettle). Gradually beat in the remaining sugar and fold into the thickened mixture. Whip half of the cream until it holds peaks. Fold in the whipped cream, and turn into the crusts. Chill several hours, or overnight. To serve, garnish with the remaining cream, whipped. Using a vegetable peeler, make chocolate curls from the chocolate bars and let drop onto the cream.


cookingFor additional recipes for feeding the masses, please check out Cooking for Crowds by Merry “Corky” White.

Princeton authors speaking at Oxford Literary Festival 2014

We are delighted that the following Princeton authors will be speaking at the Oxford Literary Festival in Oxford, UK, in the last week of March. Details of all events can be found at the links below:images5L8V7T97

Jacqueline and Simon Mitton, husband and wife popular astronomy writers and authors of From Dust to Life: The Origin and Evolution of Our Solar System and Heart of Darkness: Unraveling the Mysteries of the Invisible Universe respectively, will be speaking  on Monday 24 March at 4:00pm  http://oxfordliteraryfestival.org/literature-events/2014/Monday-24/in-search-of-our-cosmic-origins-from-the-big-bang-to-a-habitable-planet

David Edmonds, author of Would You Kill the Fat Man? The Trolley Problem and What Your Answer Tells Us  about Right and Wrong will be speaking on Monday 24 March at 6:00pm http://oxfordliteraryfestival.org/literature-events/2014/Monday-24/morality-puzzles-would-you-kill-the-fat-man

Robert Bartlett, author of Why Can the Dead Do Such Great Things? Saints and Worshippers from the Martyrs to the Reformation will be speaking on Tuesday 25 March at 2:00pm http://oxfordliteraryfestival.org/literature-events/2014/Tuesday-25/why-can-the-dead-do-such-great-things

Michael Scott, author of Delphi: A History of the Center of the Ancient World will be speaking on Wednesday 26 March at 10:00am http://oxfordliteraryfestival.org/literature-events/2014/Wednesday-26/delphi-a-history-of-the-centre-of-the-ancient-world

Simon Blackburn, author of Mirror, Mirror: The Uses and Abuses of Self-Love will be speaking on Wednesday 26 March at 4:00pm http://oxfordliteraryfestival.org/literature-events/2014/Wednesday-26/mirror-mirror-the-uses-and-abuses-of-self-love

Roger Scruton author of the forthcoming The Soul of the World will be speaking Thursday 27 March 12:00pm http://oxfordliteraryfestival.org/literature-events/2014/Thursday-27/the-soul-of-the-world

Alexander McCall Smith, author of What W. H. Auden Can Do for You will be speaking about how this poet has enriched his life and can enrich yours too on Friday 28 March at 12:00pm http://oxfordliteraryfestival.org/literature-events/2014/Friday-28/what-w-h-auden-can-do-for-youMcCallSmith_Auden

Averil Cameron, author of Byzantine Matters will be speaking on Friday 28 March at 2:00pm  http://oxfordliteraryfestival.org/literature-events/2014/Friday-28/byzantine-matters

Edmund Fawcett, author of Liberalism: The Life of an Idea will be speaking on Saturday 29 March at 10:00am http://oxfordliteraryfestival.org/literature-events/2014/Saturday-29/liberalism-the-life-of-an-idea

In addition, Ian Goldin will be giving the inaugural “Princeton Lecture” at The Oxford Literary Festival, on the themes within his forthcoming book, The Butterfly Defect: How Globalization Creates Systemic Risks, and What to Do about It on Thursday 27 March at 6:00pm  http://oxfordliteraryfestival.org/literature-events/2014/Thursday-27/the-princeton-lecture-the-butterfly-defect-how-globalisation-creates-system

 

#PiDay Activity: Using chocolate chips to calculate the value of pi

Chartier_MathTry this fun Pi Day activity this year. Mathematician Tim Chartier has a recipe that is equal parts delicious and educational. Using chocolate chips and the handy print-outs below, mathematicians of all ages can calculate the value of pi. Start with the Simple as Pi recipe, then graduate to the Death by Chocolate Pi recipe. Take it to the next level by making larger grids at home. If you try this experiment, take a picture and send it in and we’ll post it here.

Download: Simple as Pi [Word document]
Download: Death by Chocolate Pi [Word document]

For details on the math behind this experiment please read the article below which is cross-posted from Tim’s personal blog. And if you like stuff like this, please check out his new book Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing.

For more Pi Day features from Princeton University Press, please click here.


 

Chocolate Chip Pi

How can a kiss help us learn Calculus? If you sit and reflect on answers to this question, you are likely to journey down a mental road different than the one we will traverse. We will indeed use a kiss to motivate a central idea of Calculus, but it will be a Hershey kiss! In fact, we will have a small kiss, more like a peck on the cheek, as we will use white and milk chocolate chips. The math lies in how we choose which type of chip to use in our computation.

Let’s start with a simple chocolatey problem that will open a door to ideas of Calculus. A Hershey’s chocolate bar, as seen below, is 2.25 by 5.5 inches. We’ll ignore the depth of the bar and consider only a 2D projection. So, the area of the bar equals the product of 2.25 and 5.5 which is 12.375 square inches.

Note that twelve smaller rectangles comprise a Hershey bar. Suppose I eat 3 of them. How much area remains? We could find the area of each small rectangle. The total height of the bar is 2.25 inches. So, one smaller rectangle has a height of 2.25/3 = 0.75 inches. Similarly, a smaller rectangle has a width of 5.5/4 = 1.375. Thus, a rectangular piece of the bar has an area of 1.03125, which enables us to calculate the remaining uneaten bar to have an area of 9(1.03125) = 9.28125 square inches.

Let’s try another approach. Remember that the total area of the bar is 12.375. Nine of the twelve rectangular pieces remain. Therefore, 9/12ths of the bar remains. I can find the remaining area simply be computing 9/12*(12.375) = 9.28125. Notice how much easier this is than the first method. We’ll use this idea to estimate the value of π with chocolate, but this time we’ll use chocolate chips!

Let’s compute the area of a quarter circle of unit radius, which equals π/4 since the full circle has an area of π. Rather than find the exact area, let’s estimate. We’ll break our region into squares as seen below.

This is where the math enters. We will color the squares red or white. Let’s choose to color a square red if the upper right-hand corner of the square is in the shaded region and leave it white otherwise, which produces:

Notice, we could have made other choices. We could color a square red if the upper left-hand corner or even middle of the square is under the curve. Some choices will lead to more accurate estimates than others for a given curve. What choice would you make?

Again, the quarter circle had unit radius so our outer square is 1 by 1. Since eight of the 16 squares are filled, the total shaded area is 8/16.

How can such a grid of red and white squares yield an estimate of π? In the grid above, notice that 8/16 or 1/2 of the area is shaded red. This is also an approximation to the area of the quarter circle. So, 1/2 is our current approximation to π/4. So, π/4 ≈ 1/2. Solving for π we see that π ≈ 4*(1/2) = 2. Goodness, not a great estimate! Using more squares will lead to less error and a better estimate. For example, imagine using the grid below:

Where’s the chocolate? Rather than shading a square, we will place a milk chocolate chip on a square we would have colored red and a white chocolate chip on a region that would have been white. To begin, the six by six grid on the left becomes the chocolate chip mosaic we see on the right, which uses 14 white chocolate of the total 36 chips. So, our estimate of π is 2.4444. We are off by about 0.697.

Next, we move to an 11 by 11 grid of chocolate chips. If you count carefully, we use 83 milk chocolate chips of the 121 total. This gives us an estimate of 2.7438 for π, which correlates to an error of about 0.378.

Finally, with the help of public school teachers in my seminar Math through Popular Culture for the Charlotte Teachers Institute, we placed chocolate chips on a 54 by 54 grid. In the end, we used 2232 milk chocolate chips giving an estimate of 3.0617 having an error of 0.0799.

What do you notice is happening to the error as we reduce the size of the squares? Indeed, our estimates are converging to the exact area. Here lies a fundamental concept of Calculus. If we were able to construct such chocolate chip mosaics with grids of ever increasing size, then we would converge to the exact area. Said another way, as the area of the squares approaches zero, the limit of our estimates will converge to π. Keep in mind, we would need an infinite number of chocolate chips to estimate π exactly, which is a very irrational thing to do!

And finally, here is our group from the CTI seminar along with Austin Totty, a senior math major at Davidson College who helped present these ideas and lead the activity, with our chocolatey estimate for π.

Free #PiDay E-Cards from The Ultimate Quotable Einstein

Send #PiDay Greetings with these free ecards featuring Einstein’s thoughts on birthdays as found in The Ultimate Quotable Einstein, edited by Alice Calaprice.


einstein birthday 2 web



einstein birthday web



Princeton Cooks… Pumpernickel Bread

You can only work on a cookbook for so long before you want to try every single recipe in the book! To spare our blog editor the effort and calories, we invited our  Princeton colleagues to try their hand at cooking and baking the delicious treats found in Cooking for Crowds: 40th Anniversary Edition by Merry “Corky” White. This guest post is from Terri O’Prey, Associate Managing Editor at Princeton University Press, who was brave enough to try Corky’s surprising pumpernickel bread recipe. This pumpernickel actually features in the story of the smoked borscht/Julia Child rescue we posted about earlier, so I thought this was a a particularly opportune time to post this cooking demo.


Pumpernickel Bread

Terri O’Prey

 

This recipe intrigued me from the start of production. I love to bake, and I really wanted to see how the surprising (to me) ingredients—chocolate and mashed potatoes!—would play out in pumpernickel bread. I began by gathering ingredients, most of which I normally stock in my pantry (caraway seeds and rye flour were the only outliers). I usually pile everything I’ll need for a recipe haphazardly on the counter, but for this baking experiment I decided to organize myself cooking show style and premeasured everything. I enjoyed moving efficiently through the steps with each ingredient ready to go, so in the end I didn’t mind the extra dirty dishes.

01_Ingredients

First I melted the chocolate in a homemade double boiler. It became smooth and glossy, just the way melted chocolate should.

02_Step_1

03_Melted

Next I combined most of the remaining ingredients in a large bowl. The result was not beautiful, but knowing that baking transforms mixtures helped me remain optimistic.

04_TheBigMix

When I first decided to try this recipe, I wondered what to use for the very large bowl required for the next step. I don’t have a 5-gallon kettle, but my gigantic stainless steel mixing bowl stood in perfectly. (Until now, I’d never had a good reason to use it. Many thanks to my good friend Alex “All Things Kitchen” for knowing I’d need this bowl someday!) With the warm water and yeast activated, I was ready for the dough stage.

05_Yeast

First I stirred in the cornmeal mixture and rye flour and then added 3 cups of all-purpose flour until I had a soft dough.

06_WetDough 07_AddFlour 08_AddMoreFlour

09_BeforeKneading

Next came my favorite step, kneading the dough. I didn’t end up using all of the remaining flour because my dough reached the smooth, elastic stage after about 4 cups. One thing I’d change about my approach next time: use more finely mashed potatoes. Mine were on the chunky side, and while kneading I discovered some unsightly lumps (which I discarded).

10_Kneading1 11_Kneading2

I greased my clean and dry bowl with butter and readied the dough for rising.

12_ReadyToRest

I let the dough rise in the covered bowl on my sunny kitchen table. I resisted peeking because I like surprises.

13_Resting

After an hour the dough had indeed changed.

14_Rested

So I punched it down and let it rise again.

15_PunchedDown

After a half hour, I got to work dividing the dough into 3 loaves. I doubted my oven could accommodate loaves on 3 sheet pans, so I decided to make 1 large loaf and 2 smaller ones on a shared pan. The dough smelled great and was the perfect consistency for dividing. I had no trouble forming it into rustic rounds.

16_ReadyToSPlit 17_3Loaves

I let the loaves rise on covered baking sheets before brushing them with the egg wash. (Note my dog Hazel in the background. She tried assisting me but really just got in the way.)

18_EggWashedWithHazel

Next I placed the pans in the preheated oven. Knowing my oven is not precise, I set the temperature at 375°, which worked well. (I probably should invest in an in-oven gauge.) I tried not to think about the heat I was letting escape while I snapped a photo.

19_InTheOven

I checked the loaves at 50 minutes (in case my oven tinkering had gone wrong) and they weren’t quite ready. After 10 more minutes tapping produced that hollow sound, and I moved the loaves to the cooling rack.

20_FreshlyBaked

I wasn’t too concerned about slicing neatly so tasted the smallest loaf when it had cooled a bit. The pumpernickel flavor was very nice, and the caraway seeds seemed to have dissolved into the dough. Wanting to stay true to the recipe recommendations, I wrapped and refrigerated the completely cooled loaves. The next day I proudly shared my cooking with the PUP crowd by setting out hearty pumpernickel slices with butter in the office kitchen.

21_TheNextDay

 


 

Pumpernickel Bread

Pumpernickel is a bread with a secret. Some say it is prunes that distinguish it; this recipe claims it is chocolate. It will make three round loaves, which, thinly sliced, should provide appetizer portions (with chopped chicken livers, for instance) for 50.

Dough
unsweetened chocolate squares (1 oz each) 2
yellow cornmeal ¾ c
cold mashed potatoes 2 c
warm water, 115° 3½ c
molasses ¼ c
salt 2 tbs
butter or margarine 1 tbs
caraway seeds 2 tsp
active-dry-yeast packages 2
rye flour 3 c
all-purpose flour 8 c
   
Glaze  
egg yolk 1
water 3 tbs

Melt the chocolate in a double boiler over simmering water. Then, in a large bowl, combine the cornmeal, potatoes, 3 cups of the warm water, chocolate, molasses, salt, butter or margarine, and caraway seeds.

In a very large bowl (I use a 5-gallon kettle) place the remaining . cup of warm water and sprinkle on the yeast, then stir to dissolve.

Stir in the cornmeal mixture and rye flour and beat hard until well mixed. Stir in 3 cups of the all-purpose flour to make a soft dough.

Turn onto a floured board or tabletop and knead in additional flour, to 5 or more cups, to make a smooth, elastic dough. This will take about 10 minutes.

Place the dough in a greased bowl—or wash out the kettle and dry and grease it—then turn the dough to grease the top, cover, and let rise in a warm, draft-free place for about 1 hour, or until doubled in bulk. Punch down, then let rise again for 30 minutes.

Punch the dough down and turn onto a lightly floured board or table. Divide into three equal parts, shape into round loaves, and place on greased baking sheets. Cover with tea towels and let rise until double—about 45 minutes. Meanwhile, preheat the oven to 350°.

Mix the water and egg yolk for glaze and brush the loaves with the egg-yolk

liquid. Bake loaves for 1 hour, or until tapping on the bottom of the loaves produces a hollow sound. Cool thoroughly on racks, then wrap well and refrigerate. This recipe makes 3 round loaves.

NOTE: This bread slices best when one day old. It can also be successfully frozen.


This recipe is taken from:

bookjacket

Cooking for Crowds
40th Anniversary Edition
Merry White
With a new foreword by Darra Goldstein and a new introduction by the author

“[Merry White's] book, made up of recipes she collected as the caterer for the Harvard Center for European Studies, suggested a new way of entertaining, with self-serve spanakopita, petite shrimp quiche and that savior of the anxious cook, the casserole that can be made a day ahead. Edward Koren’s woolly illustrations set the tone: vegetables are our friends, and food tastes best in groups. Even though pesto and vindaloo are no longer exotic, during the holidays her attitude (and her meatballs) may be what every stressed-out host needs.”–Alexandra Lange, New York Times

Pi Day: “Was Einstein Right?” Chuck Adler on the twin paradox of relativity in science fiction

This post is extracted from Wizards, Aliens, and Starships by Charles Adler. Dr. Adler will kick off Princeton’s Pi Day festivities tonight with a talk at the Princeton Public Library starting at 7:00 PM. We hope you can join the fun!

For more Pi Day features from Princeton University Press, please click here.


Tfts56[1]Robert A. Heinlein’s novel Time for the Stars is essentially one long in-joke for physicists. The central characters of the novel are Tom and Pat Bartlett, two identical twins who can communicate with each other telepathically. In the novel, telepathy has a speed much faster than light. Linked telepaths, usually pairs of identical twins, are used to maintain communications between the starship Lewis and Clark and Earth. Tom goes on the spacecraft while Pat stays home; the ship visits a number of distant star systems, exploring and finding new Earth-like worlds. On Tom’s return, nearly seventy years have elapsed on Earth, but Tom has only aged by five.

I call this a physicist’s in-joke because Heinlein is illustrating what is referred to as the twin paradox of relativity: take two identical twins, fly one around the universe at nearly the speed of light, and leave the other at home. On the traveler’s return, he or she will be younger than the stay-at- home, even though the two started out the same age. This is because according to Einstein’s special theory of relativity, time runs at different rates in different reference frames.

This is another common theme in science fiction: the fact that time slows down when one “approaches the speed of light.” It’s a subtle issue, however, and is very easy to get wrong. In fact, Heinlein made some mistakes in his book when dealing with the subject, but more on that later. First, I want to list a few of the many books written using this theme:

  • The Forever War, by Joe W. Haldeman. This story of a long-drawn-out conflict between humanity and an alien race has starships that move at speeds near light speed to travel between “collapsars” (black holes), which are used for faster-than-light travel. Alas, this doesn’t work. The hero’s girlfriend keeps herself young for him by shuttling back and forth at near light speeds between Earth and a distant colony world.
  • Poul Anderson’s novel, Tau Zero. In this work, mentioned in the last chapter, the crew of a doomed Bussard ramship is able to explore essentially the entire universe by traveling at speeds ever closer to the speed of light.
  • The Fifth Head of Cerberus, by Gene Wolfe. In this novel an anthropologist travels from Earth to the double planets of St. Croix and St. Anne. It isn’t a big part of the novel, but the anthropologist John Marsch mentions that eighty years have passed on Earth since he left it, a large part of his choice to stay rather than return home.
  • Larris Niven’s novel A World out of Time. The rammer Jerome Corbell travels to the galactic core and back, aging some 90 years, while three million years pass on Earth.

There are many, many others, and for good reason: relativity is good for the science fiction writer because it brings the stars closer to home, at least for the astronaut venturing out to them. It’s not so simple for her stay-at-home relatives. The point is that the distance between Earth and other planets in the Solar System ranges from tens of millions of kilometers to billions of kilometers. These are large distances, to be sure, but ones that can be traversed in times ranging from a few years to a decade or so by chemical propulsion. We can imagine sending people to the planets in times commensurate with human life. If we imagine more advanced propulsion systems, the times become that much shorter.

Unfortunately, it seems there is no other intelligent life in the Solar System apart from humans, and no other habitable place apart from Earth. If we want to invoke the themes of contact or conflict with aliens or finding and settling Earth-like planets, the narratives must involve travel to other stars because there’s nothing like that close to us. But the stars are a lot farther away than the planets in the Solar System: the nearest star system to our Solar System, the triple star system Alpha Centauri, is 4.3 light-years away: that is, it is so far that it takes light 4.3 years to get from there to here, a distance of 40 trillion km. Other stars are much farther away. Our own galaxy, the group of 200 billion stars of which our Sun is a part, is a great spiral 100,000 light-years across. Other galaxies are distances of millions of light-years away.

From our best knowledge of physics today, nothing can go faster than the speed of light. That means that it takes at least 4.3 years for a traveler (I’ll call him Tom) to go from Earth to Alpha Centauri and another 4.3 years to return. But if Tom travels at a speed close to that of light, he doesn’t experience 4.3 years spent on ship; it can take only a small fraction of the time. In principle, Tom can explore the universe in his lifetime as long as he is willing to come back to a world that has aged millions or billions of years in the meantime.

 

Was Einstein Right?

This weird prediction—that clocks run more slowly when traveling close to light speed—has made many people question Einstein’s results. The weirdness isn’t limited to time dilation; there is also relativistic length contraction. A spacecraft traveling close to the speed of light shrinks in the direction of motion. The formulas are actually quite simple. Let’s say that Tom is in a spacecraft traveling along at some speed v, while Pat is standing still, watching him fly by. We’ll put Pat in a space suit floating in empty space so we don’t have to worry about the complication of gravity. Let’s say the following: Pat has a stopwatch in his hand, as does Tom. As Tom speeds by him, both start their stopwatches at the same time and Pat measures a certain amount of time on his watch (say, 10 seconds) while simultaneously watching Tom’s watch through the window of his spacecraft. If Pat measures time ∆t0 go by on his watch, he will see Tom’s watch tick through less time. Letting ∆t be the amount of time on Tom’s watch, the two times are related by the formula

where the all-important “gamma factor” is

The gamma factor is always greater than 1, meaning Pat will see less time go by on Tom’s watch than on his. Table 12.1 shows how gamma varies with velocity.

Note that this is only really appreciable for times greater than about 10% of the speed of light. The length of Tom’s ship as measured by Pat (and the length of any object in it, including Tom) shrinks in the direction of motion by the same factor.

Even though the gamma factor isn’t large for low speeds, it is still measurable. To quote Edward Purcell, “Personally, I believe in special relativity. If it were not reliable, some expensive machines around here would be in very deep trouble”. The time dilation effect has been measured directly, and is measured directly almost every second of every day in particle accelerators around the world. Unstable particles have characteristic lifetimes, after which they decay into other particles. For example, the muon is a particle with mass 206 times the mass of the electron. It is unstable and decays via the reaction

It decays with a characteristic time of 2.22 μs; this is the decay time one finds for muons generated in lab experiments. However, muons generated by cosmic ray showers in Earth’s atmosphere travel at speeds over 99% of the speed of light, and measurements on these muons show that their decay lifetime is more than seven times longer than what is measured in the lab, exactly as predicted by relativity theory. This is an experiment I did as a graduate student and our undergraduates at St. Mary’s College do as part of their third-year advanced lab course. Experiments with particles in particle accelerators show the same results: particle lifetimes are extended by the gamma factor, and no matter how much energy we put into the particles, they never travel faster than the speed of light. This is remarkable because in the highest-energy accelerators, particles end up traveling at speeds within 1 cm/s of light speed. Everything works out exactly as the theory of relativity says, to a precision of much better than 1%.

How about experiments done with real clocks? Yes, they have been done as well. The problems of doing such experiments are substantial: at speeds of a few hundred meters per second, a typical speed for an airplane, the gamma factor deviates from 1 by only about 1013. To measure the effect, you would have to run the experiment for a long time, because the accuracy of atomic clocks is only about one part in 1011 or 1012; the experiments would have to run a long time because the difference between the readings on the clocks increases with time. In the 1970s tests were performed with atomic clocks carried on two airplanes that flew around the world, which were compared to clocks remaining stationary on the ground. Einstein passed with flying colors. The one subtlety here is that you have to take the rotation of the Earth into account as part of the speed of the airplane. For this reason, two planes were used: one going around the world from East to West, the other from West to East. This may seem rather abstract, but today it is extremely important for our technology. Relativity lies at the cornerstone of a multi-billion-dollar industry, the global positioning system (GPS).

GPS determines the positions of objects on the Earth by triangulation: satellites in orbit around the Earth send radio signals with time stamps on them. By comparing the time stamps to the time on the ground, it is possible to determine the distance to the satellite, which is the speed of light multiplied by the time difference between the two. Using signals from at least four satellites and their known positions, one can triangulate a position on the ground. However, the clocks on the satellites run at different rates as clocks on the ground, in keeping with the theory of relativity. There are actually two different effects: one is relativistic time dilation owing to motion and the other is an effect we haven’t considered yet, gravitational time dilation. Gravitational time dilation means that time slows down the further you are in a gravitational potential well. On the satellites, the gravitational time dilation speeds up clock rates as compared to those on the ground, and the motion effect slows them down. The gravitational effect is twice as big as the motion effect, but both must be included to calculate the total amount by which the clock rate changes. The effect is small, only about three parts in a billion, but if relativity weren’t accounted for, the GPS system would stop functioning in less than an hour. To quote from Alfred Heick’s textbook GPS Satellite Surveying,

Relativistic effects are important in GPS surveying but fortunately can be accurately calculated. . . . [The difference in clock rates] corresponds to an increase in time of 38.3 μsec per day; the clocks in orbit appear to run faster. . . . [This effect] is corrected by adjusting the frequency of the satellite clocks in the factory before launch to 10.22999999543 MHz [from their fundamental frequency of 10.23 MHz].

This statement says two things: first, in the dry language of an engineering handbook, it is made quite clear that these relativistic effects are so commonplace that engineers routinely take them into account in a system that hundreds of millions of people use every day and that contributes billions of dollars to the world’s commerce. Second, it tells you the phenomenal accuracy of radio and microwave engineering. So the next time someone tells you that Einstein was crazy, you can quote chapter and verse back at him!

Julia Child saves the day. Corky White describes a culinary near-miss on BBC Radio 4

 

bookjacket

Cooking for Crowds
40th Anniversary Edition
Merry White
With a new foreword by Darra Goldstein and a new introduction by the author

“[Merry White's] book, made up of recipes she collected as the caterer for the Harvard Center for European Studies, suggested a new way of entertaining, with self-serve spanakopita, petite shrimp quiche and that savior of the anxious cook, the casserole that can be made a day ahead. Edward Koren’s woolly illustrations set the tone: vegetables are our friends, and food tastes best in groups. Even though pesto and vindaloo are no longer exotic, during the holidays her attitude (and her meatballs) may be what every stressed-out host needs.”–Alexandra Lange, New York Times